TIME EVOLUTION OF QUANTUM SYSTEMS RELATED TO THE QUANTUM ZENO SCHEME

At first glance it is surprising that there are papers describing experiments fitting to the quantum Zeno effect and on the other hand there are very similar experiments which seem to contradict to the quantum Zeno effect. Now the quantum Zeno effect should be related to a suppression of the time evolution in case of an “almost continuous measurement”. What does that mean? In this paper we discuss two possibilities to give a precise notion of such a suppression. That completes the considerations of the quantum Zeno scheme in our previous paper (K.-H. Fichtner and K. Inoue, 2013). AMS Subject Classification: 81P15, 46L53, 47N30


Introduction
In order to avoid technical problems and complicated calculations we restrict our considerations to the case of a finite dimensional Hilbert space H.
Let (V t := e ith ) t≥0 be a semigroup of unitary operators on H representing the time evolution of a quantum system, where h denotes a self-adjoint operator on H. ρ 0 = ρ denotes the state of the quantum system at time t 0 = 0. Furthermore we consider a sequence of time with equidistant times on the time interval [0, T ].Now, we perform measurements at time t k (k = 1, . . ., n − 1) related to the self-adjoint operator B on H and finally at time t n we perform a measurement related to the self-adjoint operator A( = B) on H.That means we will consider the following scheme we call to be the quantum Zeno scheme.
The above procedure consists of the following steps: Step 1: Starting with the state ρ 0 one performs a measurement according to B at time t 0 = 0. η 0 denotes the output of this measurement.Then according to von Neumann's rule the state ρ 0 changes into a certain state ρ0 depending on η 0 .Finally time evolution according to V T n = V t 1 −t 0 transforms the state ρ0 into the state ρ 1 representing the state of the quantum system at time t 1 .
Step 2: Starting with the state ρ 1 we repeat the procedure described above, i.e., first we perform a measurement according to B, and then apply the time evolution according to Thus as the result of step 2 we obtain the output of the measurement η 1 and a state ρ 2 at time t 2 .
Repeating the procedure again and again we obtain as the result of the Step (n-1) the value η n−1 being the output of the measurement according to B at time t n−1 , and the state ρ n of the quantum system at time t n = T depending on (η k ) n−1 k=0 .
Step n: One performs a measurement according to A at time T with output ξ.
Concerning the quantum Zeno effect, in order to make precise what suppression of time evolution could mean we will consider the following two possibilities.
Possibility 1: Let us consider measurements according to B n times without any distance of time between these measurements.In terms of our scheme one can put formally T = 0. Then there holds η k = η 0 (k = 1, . . ., n − 1).
For that reason one could relate the quantum Zeno effect in terms of our scheme to the statement That was already proved in [10].
Possibility 2: Neglecting the outputs η 0 , . . ., η n−1 of the measurements according to B our scheme leads to a transformation of the initial state ρ into a state ρ n,T,B,h at time T being the mixture of the states with respect to the distribution of the outputs (η k ) n−1 k=0 .Then one could relate the quantum Zeno effect to the statement ( But that does not hold in general.Of course, in the case of the trivial measurement related to the unity operator B = 1 H there is no suppression of the time evolution, i.e., we cannot expect that one can observe the quantum Zeno effect in the sense of possibility 2 in the general case of an operator B.Even in the case of a nontrivial measurement related to an operator B having at least one eigenspace with dimension higher than 1 one can construct counter examples.
For that reason in the following we will restrict our considerations to the case of an operator B having only one-dimensional eigenspaces.
Before starting let us mention some motivations of our work: There are certain quantum models of brain activities where the quantum Zeno effect plays an important role.For details we refer to [10,17,19].Now, let us assume that the brain acts like a quantum system.We consider the self-adjoint operator B related to EEG-measurement.Modern measuring devices are able to perform 50, 000 measurements per second.Thus in the cases of our scheme we can put T = 1 sec and n = 50, 000.Then our procedures could be considered as an "almost continuous measurement" according to B. Thus we get (cf.possibility 1) That makes statistical investigations more efficient and it fits to certain experimental experiences.
If on the other hand one would have a total suppression of the time evolution of the brain in the sense of possibility 2, then the usual activities of the brain would be stopped, i.e., the patient would die.That would contradict to the experiences.Moreover specialists are convinced that EEG-measurements do not harm the brain [21].From the mathematical point of view the reason could be that even in the case of a discretized model of EEG-measurements the structure of the eigenspaces of the corresponding operator seems to be very complicated.
Further let us mention that essential elements of our quantum model of certain brain activities are socalled self-collapses [2,3,4,5,6,7,8,9,11]. Now it is known that more than 10 11 self-collapses per second can occur and these self-collapses cause changes of a quantum state similar to the effect of measurements.The corresponding self-adjoint operator is a relatively simple one but there are higher dimensional eigenspaces.For that reason it would be interesting to check what kind of modifications of the quantum Zeno effect one could observe in case of higher dimensional eigenspaces.But that would be another problem.

On the Quantum Zeno Effect
As we have already mentioned before, in order to avoid technical problems and complicated calculations we restrict our considerations to the case of a finite-dimensional Hilbert space H with dimension d.Further we consider a self-adjoint operator B on H having only one-dimensional eigenspaces, i.e., we have the representation where (ψ k ) d k=1 represents a CONS in H, and there holds b k = b j (k = j).Now, let V = V t := e ith t≥0 be a semigroup of unitary operators on H representing the time evolution of a quantum system, where h denotes a selfadjoint operator on H.
We consider the following procedure: Fixing a state ρ of a quantum system at time t 0 = 0, T > 0, and n = 2, 3, . . .we perform measurements according to B interrupting the time evolution at times That means we consider repeated measurements according to B starting at time t 1 with equidistant times.Formally we identify the random outputs of these measurements with random numbers η 1 , . . ., η n−1 .Then using theorem 3 in [10] one easily checks Theorem 2.1.It holds Now, related to our scheme we complete the considered procedure adding a measurement according to a self-adjoint operator A at time T .Then neglecting the outputs η 1 , . . ., η n−1 of the measurements according to B leads to a linear channel C T,n acting on the set S(H) of self-adjoint operators on H, such that the expectation of the output ξ n of the measurement according to A is given by Eξ n = tr (ρC T,n (A)) .
Following the Section 2 in [10] a precise definition of the channel C T,n we obtain as follows: First we define linear channels acting on S(H) putting W j (A) := Pr ψ j APr ψ j (j = 1, . . ., d; A ∈ S(H)), Then a precise definition of C T,n is given by Now, let ρ be a state on H.Further let A be given by its unique representation i.e., (H m ) m is an orthogonal decomposition of H, and z m = z r if m = r.Again let ξ n denotes the output of a measurement according to A at time T .Using (13) we get Then using ( 12), (13), and ( 14) we obtain Thus the definition of ( 12) of the channel C T,n fits to (8).
Theorem 2.2.For each state ρ there holds Remark 2.3.In the sense of our procedure the channel is related to the trivial time evolution according to i.e., there are only repeated measurements according to B. Because of theorem 2.2 implies that That makes precise what suppression of time evolution in the sense of possibility 2 could mean.
Remark 2.4.Let Z T,n (ρ) be the state characterized by Then Z T,n (ρ) represents the state at time T in the sense of our scheme.Further, measurement according to B neglecting the output transforms the state ρ into the state W B (ρ) characterized by Then theorem 2.2 may be reformulated as follows For that reason we obtain if W B (ρ) = ρ.For instance that holds if ρ is an eigenstate of the operator B (cf. example A. Turing -already discussed in [10]).

Proof of Theorem 2.2
The following lemma represents an essential part of the proof of the theorem.
Remark 3.2.From lemma 3.1 we immediately obtain that there holds where Vt := e it ĥ (40) with the self-adjoint operator ĥ defined by ĥ The proof of theorem 2.2: Because of ( 14) that implies That completes the proof of theorem 2.2.
Remark 3.3.The proof of the theorem 2.2 shows that our procedure is related to the unitary channels C t,∞ on S(H) defined by C t,∞ := e −it ĥAe it ĥ (t ≥ 0, A ∈ S(H)). (47) One easily checks that there holds Thus we can conclude that W B (ρ) is invariant with respect to the time evolution according to V = Vt , i.e., our procedure (cf.theorem 2.2, remark 2.4) leads to a mapping ρ → W B (ρ) from the set of states into the set of V -invariant states.